Lesson 28 - Review of Right Triangle Trig & the Sine Law & Cosine Law IB Math HL1 – Santowski 4/25/2017 HL Math - Santowski
Examples – Right Triangle Trigonometry 4/25/2017 Math SL1 - Santowski
Examples – Right Triangle Trigonometry 4/25/2017 Math SL1 - Santowski
Examples – Right Triangle Trigonometry (6) Assuming that the Earth has a radius of 6380 km, determine the length of the 35th parallel. (7) To determine the width of a river, a surveyor marks a point on the bank of the river, A. Her partner is standing directly across the river from her at point C. The surveyor then walks 100 m downstream to point B, where she now has a line of sight to her partner at an angle of 58º relative to the river bank. Determine the width of the river. 4/25/2017 Math SL1 - Santowski
Examples – Right Triangle Trigonometry 4/25/2017 Math SL1 - Santowski
Examples – Right Triangle Trigonometry 4/25/2017 Math SL1 - Santowski
Examples – Right Triangle Trigonometry 4/25/2017 Math SL1 - Santowski
Examples – Right Triangle Trigonometry 4/25/2017 Math SL1 - Santowski
Examples – Right Triangle Trigonometry 4/25/2017 Math SL1 - Santowski
Examples – Right Triangle Trigonometry 4/25/2017 Math SL1 - Santowski
Examples – Right Triangle Trigonometry 4/25/2017 Math SL1 - Santowski
Examples – Right Triangle Trigonometry 4/25/2017 Math SL1 - Santowski
(A) Review of the Sine Law If we have a non right triangle, we cannot use the primary trig ratios, so we must explore new trigonometric relationships. One such relationship is called the Sine Law which states the following: 4/25/2017 HL Math - Santowski
Two acute angles, one obtuse angle If none of the angles of a triangle is a right angle, the triangle is called oblique. All angles are acute Two acute angles, one obtuse angle 4/25/2017 HL Math - Santowski
A S A ASA S A A SAA CASE 1: ASA or SAA 4/25/2017 HL Math - Santowski
S A S CASE 2: SSA 4/25/2017 HL Math - Santowski
S A S CASE 3: SAS 4/25/2017 HL Math - Santowski
S S S CASE 4: SSS 4/25/2017 HL Math - Santowski
Sine Law - Summary The law of Sines is used to solve triangles in which we have ASA, SAA and also SSA The law of Cosines is used to solve triangles in which we have SAS and SSS 4/25/2017 HL Math - Santowski
Examples 4/25/2017 HL Math - Santowski
Examples 4/25/2017 HL Math - Santowski
Examples 4/25/2017 HL Math - Santowski
(D) Examples Sine Law We can use these new trigonometric relationships in solving for unknown sides and angles in acute triangles: ex 4. Find A in ABC if a = 10.4, c = 12.8 and C = 75° ex 5. Find a in ABC if A = 84°, B = 36°, and b = 3.9 ex 6. Solve EFG if E = 82°, e = 11.8, and F = 25° There is one limitation on the Sine Law, in that it can only be applied if a side and its opposite angle is known. If not, the Sine Law cannot be used. 4/25/2017 HL Math - Santowski
(D) Examples Sine Law Mark is a landscaper who is creating a triangular planting garden. The homeowner wants the garden to have two equal sides and contain an angle of 75°. Also, the longest side of the garden must be exactly 5 m. (a) How long is the plastic edging that Mark needs to surround the garden? (b) Determine the area of the garden. 4/25/2017 HL Math - Santowski
c2 = a2 + b2 – 2 a b cos C (C) Law of Cosines: A b Have: two sides, included angle Solve for: missing side c C B a c2 = a2 + b2 – 2 a b cos C (missing side)2 = (one side)2 + (other side)2 – 2 (one side)(other side) cos(included angle) 4/25/2017 HL Math - Santowski
(C) Law of Cosines: a2 + b2 – c2 cos C = 2ab A Have: three sides Solve for: missing angle b c C B a Side Opposite Missing Angle Missing Angle a2 + b2 – c2 2ab cos C = 4/25/2017 HL Math - Santowski
(D) Cosine Law - Examples Solve this triangle a=2.4 c=5.2 b=3.5 A B C 4/25/2017 HL Math - Santowski
(D) Cosine Law - Examples 4/25/2017 HL Math - Santowski
(D) Cosine Law - Examples 4/25/2017 HL Math - Santowski
(D) Examples Cosine Law We can use these new trigonometric relationships in solving for unknown sides and angles in acute triangles: ex 1. Find c in CDE if C = 56°, d = 4.7 and e = 8.5 ex 2. Find G in GHJ if h = 5.9, g = 9.2 and j = 8.1 ex 3. Solve CDE if D = 49°, e = 3.7 and c = 5.1 4/25/2017 HL Math - Santowski
Further Mixed Practice Opportunities Nelson 10 textbook, Chap 6.1 http://mrsantowski.tripod.com/2010Math2Honors/Resources/NelsonS61p499.pdf Nelson 10 textbook, Chap 6.2 http://mrsantowski.tripod.com/2010MathSLY1/Assessments/M11SB515.pdf 4/25/2017 HL Math - Santowski